The Monty Hall Decision
- TV Program: You are a contestant. There are 3 doors, behind two of them are goats while the third has a fancy car.
- You pick a door (call it door A). You are hoping it has the car.
- Monty Hall, the game show host, opens one of the other doors (B or C) -- which has a goat. [If both doors have goats, he picks randomly.]
- Here is the game: He now allows you stick with door A (your original choice) or switch to the unopened door. What should you do?
It turns out that if you switch doors you will win the car 2/3 of the time! Why?
What is tricky is that the information which door is opened by the host (door B or door C?) does not reveal any information at all about whether or not the car is behind door A (that is still 1/3). But it does give valuable information about the remaining unopened door since opening the door with the goat has reduced the possibility that the unopened door has a goat.
To understand this problem more intuitively, think of the related exaggerated problem where there are 100 doors, 99 of which have goats while 1 has a car. You pick a door (call it "Door A"). The probability that is has the car is 1/100. The game show host opens 98 of the doors, all of which have goats. Should you stick with Door A or switch to the other door (call it "Door B")
Opening the 98 doors gives you no additional information on if your Door A has a car since you knew that 98 of the other doors had goats. However, in order for Door B to survive the drastic pruning (comparing it with each of the other 98 doors), its chance of having the car increased to 1/2. So it is smart to switch.
Door A wins when it has the car, while, because of the pruning, B wins when any of the other doors have the car.
The following table explicitly shows all of the 9 possibilities with 3 doors.
Door
You PickDoor
Has CarYou
Don't SwitchYou
Switch1 1 Win Lose 1 2 Lose Win 1 3 Lose Win 2 1 Lose Win 2 2 Win Lose 2 3 Lose Win 3 1 Lose Win 3 2 Lose Win 3 3 Win Lose Question: With the 100 door version, what is the analysis if he only opens 50 doors (all having goats)?